P780.02 Spring 2003 L7
Neutral Kaons and CP violation
Richard Kass
Since CP is not conserved in neutral kaon decay it makes more sense to use
mass (or lifetime) eigenstates rather than |k1> and |k2>:
Short lifetime state with tS  9x10-11 sec.
| k S 
Long lifetime state with tL  5x10-8 sec.
| k L 
1
1 |  |
2
1
(| k1    | k 2 ) " k  short"
(| k 2    | k1 ) " k  long"
1 |  |
 is a (small) complex number that allows for CP violation through mixing.
2
There can be two types of CP violation in kL decay:
indirect (“mixing”): kL  because of its k1 component
direct: kL because the amplitude for k2 allows k2
It turns out that both types of CP violation are present and indirect >> direct!
Review:
The strong interaction eigenstates (with definite strangeness) are: k o | k 0  and k o |k o 
If S=strangeness operator then: S | k 0 | k 0  and S | k 0   | k 0 
They are particle and anti-particle and by the CPT theorem have the same mass.
Experimentally we find:
19
(mk o  mk o ) / m  9  10
P780.02 Spring 2003 L7
Neutral Kaons and CP violation
Richard Kass
The kL and kS states are not CP (or S) eigenstates since:
CP | ks 
CP | k L 
1
CP | k1  CP | k2   
2
1 |  |
1
1 |  |
2
CP | k2  CP | k1   
In fact these states are not orthogonal!
 k S | k L  k S | k L 
1
2
| k1   | k2   | ks 
2
 | k2   | k1   | k L 
1 |  |
1
1 |  |
2 Re 
1 |  |2
If CP violation is due to mixing (“indirect”only) then the amplitude for kL is:
 k1 | k L 
1
 k1 | k2    k1 | k1   
2
1 |  |

1 |  |2
While the amplitude for kL is:
 k2 | k L 
1
 k2 | k2    k2 | k1   
2
1 |  |
1
1 |  |2
From experiments we find that || = 2.3x10-3.
However, the standard model predicts a small amount of direct CP violation too!
P780.02 Spring 2003 L7
Neutral Kaons and CP violation Richard Kass
The standard model predicts that the quantities h+- and h00 should differ very slightly
as a result of direct CP violation (CP violation in the amplitude).
Amp (k L     )
Amp (k L   o o )
h  
hoo 
 
Amp (ks    )
Amp (ks   o o )
CP violation is now described by two complex parameters,  and , with  related
to direct CP violation. The standard model estimates Re(/) to be (4-30)x10-4!
Experimentally what is measured is the ratio of branching ratios:
BR(K L      )
BR(Ks      )
2
h 
  2




1

6Re
o o
  
hoo
BR(K L    )
  2 2
BR(Ks   o o )
After many years of trying (starting in 1970’s) and some controversial experiments,
a non-zero value of Re(/) has been recently been measured (2 different experiments):
Re(/)=17.21.8x10-4
At this point, the measurement is more precise than the theoretical calculation!
Calculating Re(/) is presently one of the most challenging HEP theory projects.
P780.02 Spring 2003 L7
Richard Kass
Neutral Kaons and Strangeness Oscillations
Here we want to consider how the neutral kaon state evolves with time. This will
allow us to measure the K L, KS mass difference and the phases of the CP violation
parameters: h+- and hoo. How a two particle quantum system evolves in time is covered
in many texts including:
Particle Physics, Martin and Shaw section 10.2.5
Introduction to High Energy Physics, Perkins
Introduction to Nuclear and Particle Physics, Das and Ferbel
The Feynman Lectures, Vol III, section 11-5.
Lectures on Quantum Mechanics, Baym, Ch 2.
Note: Feynman's lectures were written in about 1963, before there was good experimental data on this
topic! For example, CP violation would not be discovered for another year.
While the following derivation is for the neutral kaon system it is
also applicable to B-meson oscillations and neutrino oscillations.
First, consider the case where CP is conserved. The CP eigenstates are |K1> and |K2>
and they are solutions to the time dependent Schrodinger equation:
  (t)
i
 Heff  (t)
t
P780.02 Spring 2003 L7
Richard Kass
Neutral Kaons and Strangeness Oscillations
i
  (t)
 Heff  (t)
t
Here Heff is a phenomenological Hamiltonian that describes the system. Since our particles
can decay, Heff is not a hermitian operator! Since there are two states in this problem
it is customary to describe Heff by a 2x2 matrix (e.g. see Das and Ferbel Chapter XII) and |K1>
and |K2> by column vectors. The Heff matrix is written in terms of the masses (m1, m2) and
lifetimes (t1, t2) of the two states:
0.5(m1  m2 )  0.25i (1 / t1  1 / t 2 )
Heff  
0.5(m2  m1 )  0.25i (1 / t 2  1 / t1)
0.5(m2  m1 )  0.25i (1/ t 2  1/ t1 )
0.5(m1  m2 )  0.25i (1/ t1  1 / t 2 )
The eigenvalues and eigenvectors of Heff are given by:
i
Heff K1(t)  (m1 
) K (t)
2 t1 1
with:
i
H eff K 2 (t )  (m2 
) K 2 (t )
2t 2
 f 1(t)


K1 (t) 
 f 1(t)
f 2 (t)


K2 (t) 
f 2 (t)
f(t) contains the time dependence
In this representation the states are orthogonal to each other: <k1(t)|k2(t)>= <k2(t)|k1(t)>=0.
The solutions to the Hamiltonian are:
K1 (t) 
i
i
 (m1 
)t
2 t1
e
K1
K2 (t)
i
i
 (m2 
)t
2t 2
e
K2
P780.02 Spring 2003 L7
Richard Kass
Neutral Kaons and Strangeness Oscillations
Consider an experiment which produces a beam of pure K0’s (at t=0) using the
strong interaction, -pK0. From the previous lecture we saw that we can express
a K0 in terms of a mixture of K1 and K2 states (and visa versa):
1
1
(| k 0   | k 0 ) | k2 
(| k 0   | k 0 )
2
2
1
1
| k 0 
(| k1   | k2 ) | k 0 
(| k1   | k2 )
2
2
Using the time dependent solutions for K1 and K2 we can find the time dependent
solution for K0:
| k1 
i
i
  i (m  i )t

 (m2 
)t
1
1
2t 1
2 t2
e
K o (t) 
K1  e
K2 


2


The amplitude for finding a K0 in the beam at a later time (t) is given by:
i
i
  i (m  i )t

 (m2 
)t
1
1
o o
2
t
2
t
1
2
K K (t)   K1  K2 ) 
K1  e
K2 
e

2


i
i
  i (m  i )t
 (m2 
)t 
1
1
2 t1
2 t2
K o K o (t)  
e
e



2


P780.02 Spring 2003 L7
Richard Kass
Neutral Kaons and Strangeness Oscillations
The probability to find a K0 in the beam at a later time is given by:
K o K o (t)
2
K o K o (t)
i
i
i
i
i
i
  i (m  i )t
 (m 2 
)t   (m1 
)t
 (m2 
)t 
1
1 
2 t1
2 t2
2 t1
2 t2
e
 e
e

e




4



2
t
i
i
i
i
i
i
i
i
  t

 (m1 
)t  (m2 
)t
 (m1 
)t  (m2 
)t 
1  t1
t2
2 t1
2 t2
2 t1
2t 2
 e
e
e
e
e
e


4


2
K o K o (t)
t
i
i
i
i
i
i
  t

(m2 m1 

)t
 (m1 m2 

)t 
1  t1
2 t1 2 t 2
2t 1 2 t 2
 e
 e t2  e
e


4


t
t 1 1  i
  t
i


(  )
2
(m
m
)t

(m1 m 2 )t 
2
1
1  t1
o o
t2
2 t1 t 2
K K (t)  e
e
e
e
e


4




o
o
K K (t)
2
t
t 1 1
  t


 (  )
1  t1
t
t2
2 t1 t 2
 e
e
 2e
cos (m2  m1 )


4


To make the units come out
right in the cos term substitute
(m2-m1)c2 for (m2-m1).
The third term in the above equation is an interference term and it causes an
oscillation that depends on the mass difference between k1 and k2.
P780.02 Spring 2003 L7
Richard Kass
Neutral Kaons and Strangeness Oscillations
We could also calculate, using the same procedure, the probability that a beam
initially consisting of K0’s contains K0’s at a later time:
K o K o (t)
2
t
t 1 1
  t


 (  )
1  t1
t

 e
 e t 2  2e 2 t1 t 2 cos (m2  m1 )


4


Note: the sign of the interference term is now “-”.
Since the strangeness of a K0 differs from a K0’s the strangeness content of the beam is
changing as a function of time. This phenomena is called strangeness oscillations.
More generally we call this phenomena flavor oscillations since it also occurs with
B-mesons (b quark oscillations) and neutrinos (e.g. nenm).
We can measure the strangeness content of
a beam as a function of time (distance) by
putting some material in the beam and counting
the number of strong interactions that have S=+1
in the final state vs. the number with S=-1.
K o p  K n S= +1
K o p  p
S= -1
Strangeness oscillations for an
initially pure K0 beam. A value
of (m2-m1)t1=0.5 is used.
P780.02 Spring 2003 L7
Richard Kass
Neutral Kaons, Flavor Oscillations, & CP Violation
CP violation requires modifying Heff to include additional complex parameters. The
details of the derivation are given in many texts, e.g. Weak Interactions of Leptons and
Quarks by Commins and Bucksbaum.
From an experimentalist's point of view a good quantity to measure is the yield (as a
function of proper time, time measured in the rest frame of the K) of +- decays from a
beam that is initially K0. Since we are measuring the sum of the square of the amplitude
|KL+-> +|KS+-> there will be an interference term in the number of ++- decays is given by:
decays/time (I(t)). The yield
of

t
t
t 1 1

I   (t)  I   (0)
e
 
 


ts
2
 h e

tL
 2 h  e
 ( 
)
2 ts t L

t
cos (m2  m1 )    



Thus by measuring this yield we gain information on the mass difference as well as the CP
violation parameters h+- and +-.
Event rate for +
decays as a function
of proper time. The best
fit requires interference
between the KL and KS
amplitudes.
5
t sec x 10-10
15
m2-m1=3.4910.009x10-6 eV
| h+- | =(2.290.01)x10-3
+- =(43.70.6)0
tS=0.893x10-10 sec
tL=0.517x10-7 sec